If `X and Y` are independent binomial vatiates `B(5,1/2) and B(7,1/2)` and the value of `P(X+Y=3)` is

Options

A. `(35)/(47)`
B. `(55)/(1024)`
C. `(220)/(512)`
D. `(11)/(204)`

Answer

B

Explanation

For X binomial variate `B(5,(1)/(2))`
`rArr p=(1)/(2), n=5,q=(1)/(2)`
For Y binomial variate `B(7,(1)/(2))`
`rArr p=(1)/(2), n=7,q=(1)/(2)`
Now, `X+Y=3`
(i) When `X=0,Y=3` , possible cases `=.^(5)C_(0)((1)/(2))^5 .^(7)C_(3)((1)/(2))^2=35((1)/(2))^12`
(ii) When `X=1,Y=2`, possible cases `=.^(5)C_(1)((1)/(2))^(5).^(7)C_(2)((1)/(2))^2=106((1)/(2))^12`
(iii) When `X=2,Y=1`, possible cases
`=.^(5)C_(2)((1)/(2))^(5) .^(7)C_(1)((1)/(2))^(7) =70 ((1)/(2))^12`
(iv) When `X=3,Y=0` possible cases `= .^(5)C_(3)((1)/(2))^(2).^(7)C_(0)((1)/(2))^2=10((1)/(2))^(12)`
`therefore` Total cases
`=((1)/(2))^(12)xx[35+105+70+10]`
`=(220)/(2^12)=(55)/(1024)` .